A trichotomy theorem for transformation groups of locally symmetric manifolds and topological rigidity

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by  Sylvain Cappell

A TRICHOTOMY THEOREM
FOR TRANSFORMATION GROUPS
OF LOCALLY SYMMETRIC MANIFOLDS
AND TOPOLOGICAL RIGIDITY
SYLVAIN CAPPELL, ALEXANDER LUBOTZKY,
6 AND SHMUEL WEINBERGER
1
0
2

n
a Abstract. LetM bealocallysymmetricirreducibleclosedman-
J
ifold of dimension ≥ 3. A result of Borel [Bo] combined with

3
Mostow rigidity imply that there exists a finite group G = G(M)

] such that any finite subgroup of Homeo+(M) is isomorphic to a
R
subgroup of G. Borel [Bo] asked if there exist M’s with G(M)
G
trivial and if the number of conjugacy classes of finite subgroups
h. of Homeo+(M) is finite. We answer both questions:
t (1) For every finite group G there exist M’s with G(M) = G,
a
m and
+
(2) the number of maximal subgroups of Homeo (M) can be
[

either one, countably many or continuum and we determine
1
(at least for dimM 6=4) when each case occurs.
v
2 Our detailed analysis of (2) also gives a complete characterization
6 of the topologicallocalrigidity and topologicalstrong rigidity (for
2
dimM 6= 4) of proper discontinuous actions of uniform lattices in
0
0 semisimple Lie groups on the associated symmetric spaces.
.
1
0
6
1
1. Introduction
:
v
i
X A positive dimensional, oriented, closed manifold M has a very
r large group of automorphisms (i.e., orientation preserving self home-
a
omorphisms). In fact this group Homeo+(M) is infinite dimensional.
But its finite subgroups are quite restricted. In 1969, Borel showed
(in a classic paper [Bo] but which appeared only in 1983 in his col-
lected works) that if M is a K(π,1)-manifold with fundamental group
Γ = π (M), whose center is trivial, then every finite transformation
1
group G in Homeo+(M) is mapped injectively into the outer automor-
phism group Out(Γ) by the natural map (or more precisely into the
subgroup Out+(Γ) – see §2 – which has an index at most 2 in Out(Γ)).
Let now M be a locally symmetric manifold of the form Γ\H/K
when H is a connected non-compact semisimple group with trivial
1
2 S. CAPPELL, A. LUBOTZKY,AND S.WEINBERGER
center and with no compact factor and Γ a torsion free uniform ir-
reducible lattice in H. In the situation in which strong rigidity holds
(i.e., if H is not locally isomorphic to SL (R)), Out(Γ) is a finite group,
2
G ≤ Out+(Γ); in fact, Out+(Γ) = N (Γ)/Γ and it acts on M as the
H
group of (orientation preserving) self isometries Isom+(M) of the Rie-
mannian manifold M. It follows now from Borel’s theorem that every
finite subgroupofHomeo+(M) is isomorphic toa subgroupof onefinite
group, G(M) = Isom+(M).
Borel ends his paper by remarking: “The author does not know
whether the finite subgroups of Homeo+(M) form finitely many con-
jugacy classes, nor whether one can find a Γ with no outer automor-
phism.”
The goal of the current paper is to answer these two questions. For
an efficient formulation of our results, let us make the following defini-
tion(s):
Definition 1.1. Let G be a finite group. An oriented manifold M will
be called G-exclusive (resp., G-weakly exclusive) if there is a faithful
action of G on M, so that G can be identified with a subgroup of
Homeo+(M) and if F is any finite subgroup of Homeo+(M), then F is
conjugate (resp., isomorphic) to a subgroup of G.
Note that Borel’s Theorem combined with strong rigidity implies
that unless H is locally isomorphic to SL (R), M as above is always
2
at least Isom+(M)-weakly exclusive.
We now claim:
Theorem 1.2. For every finite group G and every 3 ≤ n ∈ N, there
exist infinitely many oriented closed hyperbolic manifolds M = Mn(G)
of dimension n, with G ≃ Isom+(M) and when n 6= 4 these Mn(G)
are also G-exclusive.
The very special case G = {e} answers Borel’s second question
(where one can also deduce it from [BL]). Along the way it also an-
swers the question of Schultz [Sc2], attributed there to D. Burghelea,
whoaskedwhether thereexistasymmetric closedmanifoldswithdegree
onemapsontohyperbolicmanifolds. Ourexamples areevenhyperbolic
themselves.
The situation for dimension 2 is very different:
Theorem 1.3. For no group G, does there exist a G-weakly exclusive
2-dimensional closed manifold.
FINITE TRANSFORMATION GROUPS 3
In fact, for every closed, oriented surface Σ , of genus g, Homeo+(Σ)
g
has more than one (but only finitely many) isomorphism classes of
maximal finite subgroups, and this number is unbounded as a function
of g - see Proposition 6.2.
As mentioned before, for M as above with dimM ≥ 3,M is always
G-weakly exclusive. But the G-exclusiveness shown in Theorem 1.2
is not the general phenomenon. We can determine the situation in
(almost) all cases. But first we need another definition.
Definition 1.4. For an automorphism ϕ of a manifold M, denote by
Fix(ϕ) the fixed point set of ϕ and for a subgroup G ⊆ Homeo+(M),
denote its singular set S(G) = ∪{Fix(ϕ)|ϕ ∈ G,ϕ 6= id}. If M is
an oriented Riemannian manifold, then we will call S(Isom+(M)) the
singular set of M and we denote it S .
M
We note that dim(M) − dim(S ) is always even, as we are only
M
considering orientation preserving actions.
Before stating our main theorem, let us recall that in our situation,
i.e., when M is locallysymmetric, every finitesubgroup of Homeo+(M)
is contained in a maximal finite subgroup. We can now give a very
detailed answer to Borel’s first question.
Theorem 1.5 (Trichotomy Theorem). Let M = Γ\H/K a locally
symmetric manifold as above, and assume dimM 6= 2 or 4. Let
G = Isom+(M), so G ∼= N/Γ where N = N (Γ). Then one of the
H
following holds:
(a) Homeo+(M) has a unique conjugacy class of maximal finite sub-
groups, all of whose members are conjugate to Isom+(M).
(b) Homeo+(M) has countably infinite many maximal finite subgroups,
up to conjugacy or
(c) Homeo+(M) has a continuum of such subgroups (up to conjugacy).
These cases happen, if and only if the following hold, respectively:
(a) (i) S = φ, i.e., Isom+(M) acts freely on M, or
M
(ii) the singular set S is 0-dimensional and either dim(M) is di-
M
visible by 4 or all elements of order 2 act freely.
(b) M isof dimensionequal 2 (mod4), the singularsetis 0-dimensional
and some element of order 2 has a non-empty fixed point set, or
(c) the singular set S is positive dimensional, i.e., M has some non-
M
trivial isometry with a positive dimensional fixed point set.
4 S. CAPPELL, A. LUBOTZKY,AND S.WEINBERGER
The cases treated in Theorem 1.2 are with N = N (Γ) torsion free
H
(see §2), i.e., S(G) = φ where G = Isom+(M), so we are in case (a)(i)
and these manifolds M are Isom+(M)-exclusive also by Theorem 1.5.
An interesting corollary of the theorem is that if Homeo+(M) has
only finitely many conjugacy classes of maximal finite subgroups then
it has a unique one, the class of Isom+(M), in contrast to Theorem 1.3.
In dimension 4 when the action has positive dimensional singular
set, we do construct uncountably many actions. If the singular set is
finite, then we have countability, but we do not know whether/when
this countable set of actions consists of a unique possibility. As a
consequence, the following dichotomy holds in all dimensions:
Corollary 1.6. Let M be as above with arbitrary dimension. Then
Homeo+(M) has an uncountable number of conjugacy classes of finite
subgroups if and only if the singular set of Isom+(M) acting on M is
positive dimensional.
The uniqueness in Theorem 1.5 fails in the smooth case (i.e., for
Diff+(M)). In that case, the number of conjugacy classes is always
countable. The boundary between finite and infinite number of conju-
gacy classes of finite subgroups of Diff+(M) can be largely analyzed by
the methods of this paper, but works out somewhat differently (e.g.,
one has finiteness in some cases of one dimensional singular set) and
is especially more involved when the singular set is 2-dimensional. We
shall not discuss this here.
Finally, let us present our result from an additional point of view:
Given H as above and Γ a uniform lattice in it. It acts via the standard
action ρ by translation on the symmetric space H/K which topolog-
0
ically is Rd. The Farrell-Jones topological rigidity result implies that
if Γ is torsion free, every proper discontinuous (orientation preserving)
action ρ of Γ on H/K is conjugate within Homeo+(H/K) to ρ . It has
0
been known for a long time (cf. [We2] for discussion and references)
that this is not necessarily the case if Γ has torsion. Our discussion
above (with some additional ingredient based on [CDK], [We2] - see
§7) gives the essentially complete picture. But first a definition:
Definition. For Γ,H,K and ρ as above, say
0
(1) The lattice Γ has topological strong rigidity if every proper
discontinuous action ρ of Γ on H/K, is conjugate to ρ by an
0
element of Homeo+(H/K).
FINITE TRANSFORMATION GROUPS 5
(2) Γ has local topological rigidity if for every proper discontinuous
action ρ of Γ on H/K, there exists a small neighborhood U of
ρ in Hom(Γ,Homeo+(H/K)) such that any ρ′ ∈ U is conjugate
to ρ by an element of Homeo+(H/K)).
The following two results follow from Corollary 1.6 and Theorem 1.5
(see §7):
Theorem 1.7. Let H be a semisimple group, K a maximal compact
subgroup and Γ an irreducible uniform lattice in H. Then Γ satisfies
the topological local rigidity if and only if for every non-trivial element
of Γ of finite order, the fixed point set of its action on H/K is zero
dimensional.
Theorem 1.8. For H,K and Γ as in Theorem 1.7 but assuming
dim(H/K) 6= 2,4. Then one of the following holds:
(a) Γ has topological strong rigidity, i.e., it has a unique (up to conju-
gation) proper discontinuous action on H/K ≃ Rn.
(b) Γ has an infinite countable number of such actions, yet all are lo-
cally rigid.
(c) Γ has uncountably many (conjugacy classes) of such actions.
These cases happen if and only if the following hold, respectively:
(a) (i) Γ acts freely on H/K or
(ii) every torsion (i.e., non-trivial of finite order) elementof Γ has
0-dimensional fixed point set in H/K and either dim(M) ≡ 0
(mod4) or there are no elements of order 2.
(b) dim(H/K) ≡ 2(mod4), the fixed point set of every torsion element
is 0-dimensional and there is some element of order 2, or
(c) there exist a torsion element in Γ with a positive dimensional fixed
point set.
The paper is organized as follows. In §2, we prove Theorem 1.2.
In §3, we give preliminaries for the proof of Theorem 1.5, which will
be given in §4. In this proof we depend crucially on the deep works of
Farrell and Jones [FJ1] [FJ2] and Bartels and Lueck [BL] related to the
(famous) Borel conjecture as well as work of [CDK]. In §5, we analyze
manifolds of dimension 4, while in §6 we prove Theorem 1.3. Section
7 discusses topological rigidity of lattices and proves Theorem 1.7 and
1.8.
Remark. If one allows orientation reversing actions then if dimM ≥ 7
there is a trichotomy theorem; rigidity holds if the action is free or the
6 S. CAPPELL, A. LUBOTZKY,AND S.WEINBERGER
dimM ≡ 1 mod 4 or if dimM ≡ 3(4) and the elements of order 2 act
freely. The proof ofthis is similar to the one we give below forTheorem
1.5. We believe that the remaining cases, at least when dimM 6= 4,
work out similarly to that theorem.
Acknowledgment: This work was partially done while the authors
visited Yale University. They are grateful to Dan Mostow for a useful
conversation. We are also grateful to NYU, the Hebrew University,
ETH-ITS for their hospitality as well as to the NSF, ERC, ISF, Dr.
MaxRo¨ssler, theWalterHaefner FoundationandtheETHFoundation,
for their support.
2. Proof of Theorem 1.2
The proof of the theorem is in four steps:
Step I:In[BeL], M. Belolipetsky andthesecondnamed authorshowed
that for every n ≥ 3 and for every finite group G, there exist infin-
itely many closed, oriented, hyperbolic manifolds M = Mn(G) with
Isom+(M) ≃ G. More precisely, it is shown there that if Γ is the non-
0
arithmetic cocompact lattice in H = PO+(n,1) constructed in [GPS],
thenithasinfinitelymanyfiniteindexsubgroupsΓwithN (Γ)/Γ ≃ G.
H
The proof shows that Γ can be chosen so that N (Γ) is torsion free
H
and moreover N (Γ)/Γ ≃ Isom+(M) = Isom(M) for M = Γ\Hn. This
H
implies that G = N (Γ)/Γ acts on M freely, a fact we will use in Step
H
IV below.
Let M = Mn(G) be one of these manifolds, Γ = π (M). So Γ can
1
be considered as a cocompact lattice in Isom+(Hn) = PO+(n,1), the
group of orientation preserving isometries of the n-dimensional hyper-
bolic space Hn.
Step II: The Mostow Strong Rigidity Theorem [Mo] for compact hy-
perbolic manifolds asserts that if Γ and Γ are torsion free cocompact
1 2
lattices in Isom(Hn), then every group theoretical isomorphism fromΓ
1
to Γ is realized by a conjugation within Isom(Hn) (or in a geometric
2
language, homotopical equivalence of hyperbolic manifolds implies an
isomorphism as Riemannian manifolds). Applying Mostow’s theorem
for the automorphisms of Γ = π (M) implies that Aut(Γ) can be iden-
1
tified with NIsom(Hn)(Γ), the normalizer of Γ in Isom(Hn). Hence the
outer automorphism group Out(Γ) = Aut(Γ)/Inn(Γ) of Γ is isomor-
phic to NIsom(Hn)(Γ)/Γ and hence also to Isom(M), which in our case
is equal to Isom+(M) by step I.
FINITE TRANSFORMATION GROUPS 7
Step III: In [Bo], Borel showed that if Γ is a torsion free cocompact
lattice in a simple non-compact Lie group H, with a maximal compact
subgroup K and associated symmetric space X = H/K, then every
finitesubgroupF ofHomeo(M)whereM = Γ\X,ismappedinjectively
into Out(π (M)) = Out(Γ) by the natural map. If F ≤ Homeo+(M),
1
then its image is in Out+(Γ) which the kernel of the action of Out(Γ)
on Hn(Γ,Z) ≃ Z, so [Out(Γ) : Out+(Γ)] ≤ 2. Borel’s result is actually
much more general; the reader is referred to that short paper for the
generalresultandtheproofwhichusesSmiththeoryandcohomological
methods.
Anyway, applying Borel’s result for our M = Mn(G) finishes the
proof of the first part of Theorem 1.2. In particular, one sees that in
all these examples, G = Isom+(M) acts freely on M since the isometry
group, in this case, is the group of covering transformations which act
freely on M.
Step IV: We have shown so far that whenever a finite group F acts
on M as above, there is a natural injective homomorphism F ֒→
Out+(π (M)) ∼= Isom+(M). Denotethe image ofF inIsom+(M) by L.
1
Our next goal is to show that F is conjugate to L within Homeo+(M).
For ease of reading we will call M with the action of F, M′, to avoid
confusion.
There is actually an equivariant map M′ → M that is a homotopy
equivalence. To see this, note that N (Γ) is torsion free and hence
H
so is Γ, the preimage of L in N (Γ) w.r.t. the natural projection
H
N (Γ) → Out(Γ) = Out(π (M)). Similarly, let us consider all of the
H 1
possible lifts of all of the elements of F to the universal cover, which
form a group Γ′ (the orbifold fundamental group of M′/F, which we
presently show is the genuine fundamental group) that fits in an exact
sequence:
′
1 → Γ(= π (M)) → Γ → F → 1
1
As Γ is centerless and F and L induce the same outer automorphism
′ ′
group, it follows that Γ is also torsion free and as a corollary F = Γ/Γ
acts freely on M′. Hence M′/F is homotopy equivalent to M/L as
′
both have Γ ≃ Γ as their fundamental group. By the Borel conjec-
ture for hyperbolic closed manifolds (which is a Theorem of Farrell
and Jones [FJ1] for n ≥ 5 and of Gabai-Meyerhoff-Thurston [GMT]
for n = 3) the map M′/F → M/L is homotopic to a homeomorphism
which preserves π (M′) = π (M), as did the original homotopy equiv-
1 1
alence. Since liftability in a covering space is a homotopy condition,
8 S. CAPPELL, A. LUBOTZKY,AND S.WEINBERGER
this homeomorphism can be lifted to the cover M′ → M, producing a
conjugating homeomorphism between the actions. Theorem 1.2 is now
proved.
In summary, the above proof is analogous to (and relies on) the fact
that Mostow rigidity gives a uniqueness of the isometric action (or
in different terminology, the uniqueness of the Nielsen realization of
a subgroup of Out(Γ)). At the same time, the Farrell-Jones/Gabai-
Meyerhoff-Thurston rigidity gives the uniqueness of the topological re-
alization in the case of free actions. We will see later that this freeness
condition is essential.
3. Some ingredients for the proof of Theorem 1.5
The proof of Theorem 1.5 is based on results, sometimes deep theo-
rems, some of which are well-known and others which might be folklore
(or new). We present them in this section and use them in the next
one.
Ingredient 3.1. For v ≥ 3, there exists infinitely many non-simply
connected homology spheres Σv, each bounding a contractible manifold
Xv+1 such that the different fundamental groups π (Σ) are all freely
1
indecomposable and are non isomorphic to each other. Moreover, X ×
[0,1] is a ball.
Proof. For v > 4 this is very straightforward. According to Kervaire
[Ker], a group π is the fundamental group of a (PL) homology sphere
iff it is finitely presented and superperfect (i.e., H (π) = H (π) =
1 2
0). Moreover every PL homology sphere bounds a PL contractible
manifold (this is true for v ≥ 4, and for v = 3 in the topological
category [Fr]). The product of a contractible manifold with [0,1] is
a ball as an immediate application of the h-cobordism theorem (see
[Mi1]).
For v = 3, we could rely on the work of Mazur [Ma] in the PL
category, but would then need to use subsequent work on the structure
of manifolds obtained by surgery on knots. Instead, as we will be
workinginthetopologicalcategory,werelyon[Fr]whichshowsthatthe
analogueofalloftheaboveholdstopologicallyforv = 3, asidefromthe
characterization of fundamental groups: however, using the uniqueness
of the JSJ ([JS], [J]) decomposition of Haken 3-manifolds, homology
spheres obtained by gluing together nontrivial knot complements are
trivially distinguished from one another.
FINITE TRANSFORMATION GROUPS 9
For v = 4, note that all the fundamental groups of the v = 3 case
arise here as well: if Σ3 is a homology sphere then ∂(Σo×D2) is a ho-
mology 4-sphere with the same fundamental group (where Σo denotes,
(cid:3)
as usual, the punctured manifold).
We also need:
Ingredient 3.2. For m−1 > c ≥ 3 and every orientation preserving
0
linear free action ρ of G = Z on Sc0 (in particular, c is odd), there
p 0
exist an infinite number of homology spheres Σc0 with non-isomorphic
fundamental groups and with a G = Z -free action satisfying: For each
p
such Σ there exists an action of G on Bm fixing 0 ∈ Bm such that
(1) The action of G on Sm−1 is isomorphic to the linear action ρ⊕
Identity, and
(2) the local fundamental group πlocal(Bm \ F,0) is isomorphic to
1
π (Σ), when F is the fixed point set. Moreover, this action is
1
topologically conjugate to a PL action on a polyhedron.
Let us recall what is meant by the local fundamental group: This
is the inverse limit lim π (U ,x ) where the {U } is a sequence of
←− 1 α α α
α
connected open neighborhoods converging down to 0, and x ∈ U \F
α α
is a sequence of base points. Note that by the Jordan Curve Theorem,
U \F is connected as codim F ≥ 2. Also the induced maps are well
α
defined up to conjugacy, so the limit is well defined.
Proof. For every homology sphere Σ′ of odd dimension c , let Σ =
0
pΣ′ = Σ′#Σ′#···#Σ′ p times. We now give Σ a free Z action, by
p
taking connected sum along an orbit of the free linear action on Sc0
with the permutation action on pΣ′. The action on Sc0 bounds a linear
disk Dc0+1. One can take the (equivariant) boundary connect sum of
this disk with pX,X the contractible manifold that Σ′ bounds to get
a contractible manifold Z with Z action fixing just one point which is
p
locally smooth at that point and has the given local representation ρ
there.
For motivation, consider now c(Σ)×Bm−c0−1 where c(Σ) is the cone
of Σ. It is a ball, by Edwards’s theorem [D] (combined with the h-
cobordism theorem), and has an obvious Z action as desired except
p
that the action on the boundary is not linear: the fixed set is Sm−c0−2
but it is not locally flat.
Instead, let Z be the locally linear contractible manifold bounded by
Σ, constructed above. The manifold Z∪(Σ×[0,1])∪Z is a sphere (by
10 S. CAPPELL, A. LUBOTZKY,AND S.WEINBERGER
the Poincar´e conjecture). If one maps this to [0,1] by the projection on
Σ×[0,1] and extending by constant maps on the two copies of Z, then
the mapping cylinder of this map ϕ : Z ∪ (Σ × [0,1]) ∪ Z → [0,1] is
a manifold, again by Edwards’s theorem. It has an obvious Z action
p
with fixed set an interval. The action on the boundary sphere is locally
smooth with two fixed points, so that an old argument of Stallings
[St2] shows that it is topologically linear with ρ as above. Note that
the nonlocally flat points of the fixed point set correspond to the points
where the local structures is c(Σ)×[0,1]; hence the local fundamental
group is π (Σ), as required. This proves the result for the case m =
1
c +2.
0
For m > c +2, one can spin this picture: Map (Sm−c0−1×Zc0+1)∪
0
(Bm−c0−1 × Σ) to Bm−c0 in the obvious way and again the mapping
cylinder producesaball withlocallylinearboundariesanddesiredfixed
set. This time the linearity of the boundary action follows from Illman
(cid:3)
[I].
We will also need the following group theoretical result:
Proposition 3.3. If {π }∞ and {π′}∞ are two infinite countable
i i=1 i i=1
∞
families of non-isomorphic freely indecomposable groups such that ∗ π
i
i=1
∞
is isomorphic to ∗ π′, then after reordering for every i, π is isomor-
i i
i=1
phic to π′.
i
Proof. Recall that by the Bass-Serre theory, a group Γ is a free product
∞
∗ π if and only if Γ acts on a tree T with trivial edge stabilizers and a
i
i=1
contractible quotient and with one to one correspondence between the
vertices of T and the conjugates of π (i ∈ N) in Γ, where each vertex
i
∞ ∞
corresponds toitsstabilizer. Nowassume Γ ≃ ∗ π andalso Γ ≃ ∗ π′
i j
i=1 j=1
with the corresponding trees T and T′. Fix i ∈ N, as Γ acts on T′ with
trivial edge stabilizers and π is freely indecomposable, π fixes a vertex
i i
of T′. Hence there exists j ∈ N s.t. π ⊆ π′τ. In the same way π′τ is a
i j j
subgroup of some πδ for some δ ∈ Γ. This means that π ⊆ πδ. But in
k i k
a free product a free factor cannot have a non-trivial intersection with
another factor or with a conjugate of it. Moreover, if π ∩ πδ 6= {e},
i i
then πδ = π . Indeed, if g is in this intersection, it fixes the fixed vertex
i i
of π as well as that of πδ, hence also the geodesic between them, in
i i
contradiction to the fact that Γ acts with trivial edge stabilizers.
We deduce that π ⊆ π′τ ⊆ π and hence π = π′τ.
i j i i j